A hyperspectral image is a three dimensional array of numbers {xi, yj, λk}, usually called a “cube”, consisting of the intensities of light observed at each of a discrete set of wavelengths, and at every spatial pixel in a scene. A cube can be visualized as a stack of single band images of the spatial scene, with each layer representing a different wavelength band. FIG. 1 illustrates an example of a hyperspectral cube 100, which can be collected using a line scanner 102 such as the Spatially Enhanced Broadband Array Spectrograph System (SEBASS) sensor from The Aerospace Corporation of El Segundo, Calif. In this example, movement of the aircraft (or other object equipped with the line scanner 102) effects a ground path scan along a direction indicated by flight line arrow 104. By way of example, the line scanner 102 includes an array of Long Wave Infra-Red (LWIR) sensors that provide 128 LWIR bands (e.g., 7.8 to 13.5 μm, Δλ-0.04 μm) perpendicular to 128 spatial pixels, thus providing a series of sensor frames along the ground path scan direction.
Hyperspectral imaging (HSI) offers an extremely powerful tool for detecting solid targets or gaseous constituents in a complex background or mixture. The power of a hyperspectral system comes from the large number of individual wavelength bands in the spectrum of light collected at each “pixel” (spatial location) in a scene. By facilitating direct observation of spectral features at high resolution, over a broad range of wavelengths, hyperspectral imaging makes it possible to detect and identify many different materials, gases, etc., with a single system. In general, the spectral trace of a spatial pixel in a hyperspectral data cube will be influenced by several factors. These include emission from the target and atmosphere, absorption of light by the atmosphere or intervening gases, various atmospheric scattering effects, and, in the ultraviolet (UV) and visible through mid-wave infrared (MWIR) regimes, the characteristics of the light sources illuminating the scene. The light collected from a pixel may be dominated by a single object or substance, or by multiple objects, as in the case where the spatial resolution is insufficient to fully separate objects, or when light emitted by objects on the ground passes through a gas plume. Thus a given observed spectrum can be a complex mixture of characteristics of several different constituents. “Analysis” or “demixing” of hyperspectral data generally means detecting the presence and perhaps estimating the concentration of one or more specific objects or substances, by recognizing signatures of these substances in the spectral data. To this end, it would be useful to be able to suppress the influences of other entities (clutter) that may be mixed with the signature of interest in the spectrum collected at a given spatial location.
The most common approach to detection and quantification of pre-specified target signatures in mixed spectra involves use of a matched filter. A matched filter can be thought of as a mathematical operation performed on a data set that maximizes the influence of the desired signature in the output stream, while minimizing the influence of background clutter and noise. In the hyperspectral context, a matched filter is typically a high dimensional vector (denoted by F) that is approximately orthogonal to all the vectors that represent background and clutter signatures in the scene of interest, while having a significant projection on the particular target signature to which it is matched. Typically, the filter is scaled so that the dot product F·T=1, where T is a unit vector parallel to the target spectrum. Thus the filter output (i.e., the dot product of the filter with an observed spectral vector) can be used to infer the magnitude of the contribution of the target signature in the observed spectrum. If multiple targets are specified, each filter vector should also be orthogonal to all the other target signatures. Matched filters are derived via an optimization process. There are several approaches to deriving matched filters, depending on the details of the way the problem is posed, and what kind of a priori knowledge of the scene is available. The equations below illustrate a simple example of the optimization process, appropriate when there is no advance knowledge of the background, when there is a single target signature T, and when it is possible to select “training” spectra that contain all of the important background signatures but no target-bearing signatures:
Find F that minimizes the objective function C
                    C        =                                            ∑              i                        ⁢                                          (                                  F                  ·                                      d                                          i                      ⁢                                                                                          ⁢                      training                      ⁢                                                                                          ⁢                      set                                                                      )                            2                                +                      λ            ⁡                          [                                                F                  ·                  T                                -                1                            ]                                                          (        1        )            F∝M−T
Where λ is a Lagrange multiplier that adjoins the constraint F·T=1, {di training set} is the “training set”, and M is the covariance matrix for the training set. The optimization described by equation (1) produces a filter vector F that is as nearly orthogonal to all the vectors {dj} in the training set as possible, subject to the constraint condition, which enforces the normalization condition mentioned above. In this example, optimization is a two-step process. First, a tractable mathematical idealization of the problem to be solved is constructed (step 1); then a good solution to the equations that embody the idealization is found (step 2). The phrase “Optimal Solution” usually means that step 2 has been done perfectly, but the quality of such a solution is limited by the error made in step 1. Better idealizations lead to better “optimal” solutions. The standard matched filter is optimal in the sense of step 2, but the mathematical idealization on which it is based has weaknesses. In fact, the use of the quadratic objective function C that defines the standard matched filter idealization (equation 1 and its more sophisticated analogues in conventional matched filter theory) is motivated primarily by considerations of mathematical convenience, i.e., by the fact that quadratic objective functions lead directly to linear systems of equations that are easy to solve, rather than a belief that this is the best possible representation of the problem.
As in any filter process, matched filters computed via equation (1) or similar equations are subject to false alarms and limitations on their ability to detect very weak targets (sensitivity). High false alarm rates can pose a serious threat to a sensor system: they reduce the user's confidence in the products of the system, and may also threaten its economic viability. The costs of the resources expended in response to a detection (e.g., firing an expensive missile, sending out a ground crew, etc.) are increased, sometimes dramatically, by the presence of false alarms. For this reason, analysts are usually employed to study and attempt to verify candidate detections when critical decisions must be based on these, but this process is also costly (again the cost scales with the false alarm rate), error-prone, and in many cases there is insufficient time to do it well, because HSI systems typically generate data at very high rates. For example, airborne HSI systems currently in operation may accumulate data at rates of the order of a gigabyte per minute or more. For applications that involve scanning a single site of modest size at a known location, or a small set of such sites, man-in-the-loop analysis and rejection of multiple false alarms is not an unduly burdensome problem, when it can be done accurately. On the other hand, applications involving wide area searches, e.g., military reconnaissance operations, search and rescue operations, natural resource surveys, etc., may require continuous collection over periods of many hours per day, for many days. In these cases, the task of analyzing the collected data and returning results in a timely fashion becomes very substantial, especially if a human analyst must guide the process, and a high false alarm rate may result in an intolerable operational burden, and in significantly reduced system reliability.